Pauli Approximations to the Self-Adjoint Extensions of the Aharonov-Bohm Hamiltonian

It is well known that the formal Aharonov-Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can be considered as limits of regularised Aharonov-Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of non-zero diameter. We show that not all the self-adjoint extensions in this two-parameter family can be obtained by these approximations, but only two one-parameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding self-adjoint extension. aDepartment of Mathematical Physics, University College Dublin (National University of Ireland, Dublin), Belfield, Dublin 4, Ireland. be-mail: [email protected] con leave of absence from Department of Mathematics, University of Malta, Msida MSD 06, Malta. de-mail: [email protected] eResearch Associate, School of Theoretical Physics, Dublin Institute for Advanced Studies. Pauli Approximations to the Aharonov-Bohm Hamiltonian 2